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Full text of "Treatise On Analysis Vol-Ii"

9   THE PLANCHEREL-GODEMENT THEOREM       379

an isomorphism T0 of the prehilbert space A/ng onto its image in #2(8^)
such that T0 - 7ig(x) = x, it is enough to show that this image is dense in
?(Sff) n ^(Sg,mg)y with respect to the topology of&l(Sg,mg). Given e > 0
and any function F e JTC(S^)3 it is sufficient to prove the existence of x e A
such that

I

Now, we have seen earlier that there exists a function ti = %xixi (xteA)

i

which does not vanish on Supp(F), so that we may write F = Gfi, where
G 6 ?TC(S0). By virtue of (ii), there exists y e A such that

for all x e Sg > and it follows that

(W) - fl(*)Kx)l2 dmg(x)  (8/N2(fl))2 f |

Finally, for x and y in A, we have by definition

 71,60) = T0  ng(xy) = *j> = M(x)  (T0 - ng

and by extending T0 by continuity to an isomorphism Tof Hg onto Lc(S^, mg)
we have TUg(x) = M(x)T. The equality of the norm of 1  lHg + Ug(x) in
^(VLg) and the norm of 1 + x in ^C(S) is a consequence of the Gelfand-
Neumark theorem (15.4.14).

(15.9.2.22)    Uniqueness of Sg and mg.

It is immediately checked that the hermitian forrn^' satisfies (15.6.3) and
(15.7.3), hence is a bitrace on A. Since the functions x are bounded, we have

g'(xy, xy) = [ \x(X)\2\P(x)\2 dmtf)  \\x\\2g'(y, y\

J

so that g' satisfies the condition (U). To prove that g' satisfies (N), let S'
denote the closure of S in H(A) u {0}, so that S' = S if S is compact, and
S' = S u {0} otherwise. By virtue of (13.11.6), it is enough to show that the
functions A + xj>, where A G C and x, y e A, form a total set in the space
<^C(S'). The proof is the same as in (15.9.2.11), using the Stone-Weierstrass
theorem and the fact that by definition the functions x separate the points of
S u {0}. The ideal ng, is the set of all x e A for which x is /w-negligible; but
since the support of m is S, and x is continuous, x can be m-negligible only if
x = 0, and therefore ng>(x) H- x is an isomorphism of A/% onto the algebra of
the functions x such that (ng,(x) \ ng,(yj) = \x(y)$(l) dm(%). The argument of
(15.9.2.21) then shows that the set of functions x is dense in JSfc(S, m), and and therefore definesions xnxn therefore converge uniformly on S^ to G, and hence,